Showing papers by "Zhong-Zhi Bai published in 2004"

Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems

Abstract: For the positive semidefinite system of linear equations of a block two-by-two structure, by making use of the Hermitian/skew-Hermitian splitting iteration technique we establish a class of preconditioned Hermitian/skew-Hermitian splitting iteration methods. Theoretical analysis shows that the new method converges unconditionally to the unique solution of the linear system. Moreover, the optimal choice of the involved iteration parameter and the corresponding asymptotic convergence rate are computed exactly. Numerical examples further confirm the correctness of the theory and the effectiveness of the method.

A Class of Nested Iteration Schemes for Linear Systems with a Coefficient Matrix with a Dominant Positive Definite Symmetric Part

Abstract: We present a class of nested iteration schemes for solving large sparse systems of linear equations with a coefficient matrix with a dominant symmetric positive definite part These new schemes are actually inner/outer iterations, which employ the classical conjugate gradient method as inner iteration to approximate each outer iterate, while each outer iteration is induced by a convergent and symmetric positive definite splitting of the coefficient matrix Convergence properties of the new schemes are studied in depth, possible choices of the inner iteration steps are discussed in detail, and numerical examples from the finite-difference discretization of a second-order partial differential equation are used to further examine the effectiveness and robustness of the new schemes over GMRES and its preconditioned variant Also, we show that the new schemes are, at least, comparable to the variable-step generalized conjugate gradient method and its preconditioned variant

Skew-Hermitian triangular splitting iteration methods for non-Hermitian positive definite linear systems of strong Skew-Hermitian parts

Abstract: By further generalizing the skew-symmetric triangular splitting iteration method studied by Krukier, Chikina and Belokon (Applied Numerical Mathematics, 41 (2002), pp. 89–105), in this paper, we present a new iteration scheme, called the modified skew-Hermitian triangular splitting iteration method, for solving the strongly non-Hermitian systems of linear equations with positive definite coefficient matrices. We discuss the convergence property and the optimal parameters of this new method in depth. Moreover, when it is applied to precondition the Krylov subspace methods like GMRES, the preconditioning property of the modified skew-Hermitian triangular splitting iteration is analyzed in detail. Numerical results show that, as both solver and preconditioner, the modified skew-Hermitian triangular splitting iteration method is very effective for solving large sparse positive definite systems of linear equations of strong skew-Hermitian parts.

Combinative preconditioners of modified incomplete cholesky factorization and Sherman-Morrison-Woodbury update for self-adjoint elliptic Dirichlet-periodic boundary value problems

Abstract: For the system of linear equations arising from discretization of the second-order selfadjoint elliptic Dirichlet-periodic boundary value problems, by making use of the special structure of the coefficient matrix we present a class of combinative preconditioners which are technical combinations of modified incomplete Cholesky factorizations and ShermanMorrison-Woodbury update. Theoretical analyses show that the condition numbers of the preconditioned matrices can be reduced to O(h−1), one order smaller than the condition number O(h−2) of the original matrix. Numerical implementations show that the resulting preconditioned conjugate gradient methods are feasible, robust and efficient for solving this class of linear systems. Mathematics subject classification: 65F10, 65F50.